I think the question we now have to ask to resolve the remaining confusion is—what, exactly, is it that Beauty is uncertain about, and at what time?
The variables we are considering only seem to make sense if Beauty is having woken up as part of the experiment. That is, assuming x means “the coin came up heads or tails”, y means “it is Monday or Tuesday”, and z means “I am awake or asleep”—i.e., we’re dealing with uncertainty about facts that are already fixed, just unknown. Then these do not make sense outside that context.
Using that interpretation, then, and sticking to that context, we get the answer of 1⁄2, as if Beauty has just been woken up, she cannot allocate any probability mass to the possibility that she is asleep.
What other interpretations could there be? Perhaps the coin has not yet been flipped, and x is “the coin will come up heads (tails)”, y is “it will be Monday (Tuesday) when I wake up”, z is “I will be awake (asleep) when I wake up” (!). Of course, if the coin has not yet been flipped, I think we can agree 1⁄2 has to be the right answer. (Which then leads to the argument that it has to be 1⁄2 as she hasn’t gained any information, but I guess that’s been gone over before.) But the problem is that this y doesn’t seem well-defined, as she might be woken up more than once. (Hm, this is sounding familiar as well...) We could perhaps introduce separate variables for being woken up on each day; from the pre-flip point of view, that makes more sense. But it still gets you an answer of 1⁄2.
This is all I can come up with; I’m not seeing what other interpretations there could be. Could someone explain just what ‘x’, ‘y’, and ‘z’ correspond to—if they do correspond to anything well-defined rather than having to be thrown out—in the interpretations that get you 1/3? I don’t see any way for the probabilities to represent her uncertainty at the time of waking, while still having her assign nonzero probability to the possibility that she’s asleep.
I think the question we now have to ask to resolve the remaining confusion is—what, exactly, is it that Beauty is uncertain about, and at what time?
“At what time” doesn’t matter in this formalism. You can be uncertain about future events or about past events, all that matters is how you update your uncertainty upon receiving new information.
So a triplet (x,y,z) represents, in the abstract, a conceivable configuration of the component uncertainties in the experimental setup. The coin could have come up heads or tails; it could be Monday or Tuesday; Beauty can be woken up on that day, or left asleep.
The joint probability P(x,y,z) is the plausibility we assign—in a timeless manner—to the corresponding propositions. Strictly speaking, it should be P(x,y,z|B) where B is our background information about the experiment: the rules, the fact that the coin is unbiased (or not known to be biased), and so on.
Our background information directs how we allocate probability mass to the various points in the sample space: P(T,T,S) corresponds to “the coin comes up tails, the day is Tuesday, Beauty is asleep”. The rules of the experiment require that this be zero.
On the other hand, P(H,T,S) corresponds to “the coin comes up heads, the day is Tuesday, Beauty is asleep”, and this can be non-zero.
When you learn (“condition on”) some new information, the probability distribution is altered: you only keep the points which correspond to this particular variable having the value(s) you learned, and you renormalize so that the total probability is 1. So, on learning “heads” you keep only the points having x=H. On learning what day it is you keep only the points having that value for y.
When Beauty wakes up, she learns the value of z, so she can condition on z. That means she throws away the part of the joint distribution where she was supposed to be asleep. If that part of the joint distribution did contain some probability mass (as I’ve argued above it can), then that can make P(x|z=W) something other than 1⁄2.
OK, this seems quite helpful.
I think the question we now have to ask to resolve the remaining confusion is—what, exactly, is it that Beauty is uncertain about, and at what time?
The variables we are considering only seem to make sense if Beauty is having woken up as part of the experiment. That is, assuming x means “the coin came up heads or tails”, y means “it is Monday or Tuesday”, and z means “I am awake or asleep”—i.e., we’re dealing with uncertainty about facts that are already fixed, just unknown. Then these do not make sense outside that context.
Using that interpretation, then, and sticking to that context, we get the answer of 1⁄2, as if Beauty has just been woken up, she cannot allocate any probability mass to the possibility that she is asleep.
What other interpretations could there be? Perhaps the coin has not yet been flipped, and x is “the coin will come up heads (tails)”, y is “it will be Monday (Tuesday) when I wake up”, z is “I will be awake (asleep) when I wake up” (!). Of course, if the coin has not yet been flipped, I think we can agree 1⁄2 has to be the right answer. (Which then leads to the argument that it has to be 1⁄2 as she hasn’t gained any information, but I guess that’s been gone over before.) But the problem is that this y doesn’t seem well-defined, as she might be woken up more than once. (Hm, this is sounding familiar as well...) We could perhaps introduce separate variables for being woken up on each day; from the pre-flip point of view, that makes more sense. But it still gets you an answer of 1⁄2.
This is all I can come up with; I’m not seeing what other interpretations there could be. Could someone explain just what ‘x’, ‘y’, and ‘z’ correspond to—if they do correspond to anything well-defined rather than having to be thrown out—in the interpretations that get you 1/3? I don’t see any way for the probabilities to represent her uncertainty at the time of waking, while still having her assign nonzero probability to the possibility that she’s asleep.
“At what time” doesn’t matter in this formalism. You can be uncertain about future events or about past events, all that matters is how you update your uncertainty upon receiving new information.
So a triplet (x,y,z) represents, in the abstract, a conceivable configuration of the component uncertainties in the experimental setup. The coin could have come up heads or tails; it could be Monday or Tuesday; Beauty can be woken up on that day, or left asleep.
The joint probability P(x,y,z) is the plausibility we assign—in a timeless manner—to the corresponding propositions. Strictly speaking, it should be P(x,y,z|B) where B is our background information about the experiment: the rules, the fact that the coin is unbiased (or not known to be biased), and so on.
Our background information directs how we allocate probability mass to the various points in the sample space: P(T,T,S) corresponds to “the coin comes up tails, the day is Tuesday, Beauty is asleep”. The rules of the experiment require that this be zero.
On the other hand, P(H,T,S) corresponds to “the coin comes up heads, the day is Tuesday, Beauty is asleep”, and this can be non-zero.
When you learn (“condition on”) some new information, the probability distribution is altered: you only keep the points which correspond to this particular variable having the value(s) you learned, and you renormalize so that the total probability is 1. So, on learning “heads” you keep only the points having x=H. On learning what day it is you keep only the points having that value for y.
When Beauty wakes up, she learns the value of z, so she can condition on z. That means she throws away the part of the joint distribution where she was supposed to be asleep. If that part of the joint distribution did contain some probability mass (as I’ve argued above it can), then that can make P(x|z=W) something other than 1⁄2.